Algorithmic properties of structures

Abstract

To formulate properties of mathematical structures such as the system of natural numbers or plane Euclidean geometry, we have at our disposition a vast array of different formal languages. By far the best unders tood such language is that of first-order predicate logic. Other languages have been in t roduced with the aim to increase or to specialize its expressive power. In the present paper there arises another language I whose aim is to formulate those properties that are verifiable by experiments in the structure. By this we mean roughly the following. We are given a structure in which some operations and decisions can be per fo rmed in an effective way (either directly within the structure or in some roundabout way by translation to the integers as in Malcev [5] or Rabin [7]). As a first step we invent an algorithmic language in which we can formulate programs 1I based on these operations and'decisions. The carrying out of such a program II then stands for an exper iment with the given structure. I f the exper iment is successful, it verifies a certain property of the structure. It is convenient to normalize the notion of success to that of termination. Thus we are interested in formulat ing for each program II a s tatement ~0 such that 11 terminates in a structure 2[ if and only if 9 expresses a property of 2[. Various notions of constructive verifiability have been proposed in the literature, mainly in connection with a proposed formalization of the intuitionistic point of view (e.g., Kleene and Vesley [2], Kreisel [3], [4]). The most important difference in our approach lies in the fact that these authors choose the language in which the verifiable statements are formulated first and invent afterwards a more or less intuitive notion of verifiability to fit the language. In this light, what is obtained here may be viewed as a formalist reconstruction of finitism rather than of intuitionism.

DOI: 10.1007/BF01705528

Cite this paper

@article{Engeler1967AlgorithmicPO, title={Algorithmic properties of structures}, author={Erwin Engeler}, journal={Mathematical systems theory}, year={1967}, volume={1}, pages={183-195} }